A stabilized discontinuous finite element method for elliptic problems

A new finite element method is proposed and analysed for second order elliptic equations using discontinuous piecewise polynomials on a finite element partition consisting of general polygons. The new method is based on a stabilization of the well-known primal hybrid formulation by using some least-squares forms imposed on the boundary of each element. Two finite element schemes are presented. The first one is a non-symmetric formulation and is absolutely stable in the sense that no parameter selection is necessary for the scheme to converge. The second one is a symmetric formulation, but is conditionally stable in that a parameter has to be selected in order to have an optimal order of convergence. Optimal-order error estimates in some H1-equivalence norms are established for the proposed discontinuous finite element methods. For the symmetric formulation, an optimal-order error estimate is also derived in the L2 norm. The new method features a finite element partition consisting of general polygons as opposed to triangles or quadrilaterals in the standard finite element Galerkin method. Copyright © 2002 John Wiley & Sons, Ltd.

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